If your company has the means invest in a zamboni type floor cleaning unit. If the company doesn't want to put up that kind of money rent a unit. Some rental companies can even provide some expertise in the unit and cleaning agent that would work best in your application.
I will tell you, we use a zamboni type unit which has some ozone generator on it. The cleaning is OK, not great. Areas that are really dirty we pre-spray chlorine solution on that area before the zamboni unit drives over it.
There's no splashing, or residual water leftover. Just some wetness on the ground which dries in a matter of minutes.
A cheap method is stiff bristle brush. Spray chlorine solution on the floor, brush, and follow it with a mop. You don't want to be using a lot of water prior or with the brushing to keep splash and splatter minimized. You will have to clean or replace the brush head and mop frequently with this method.